Characterizations of some classes of finite soluble groups

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group G is supersoluble if and only if G has a normal subgroup D such that G/D is supersoluble and D avoids every chief factor of G between \(V^{G}\) $V^G$ and \(V_{G}\) $V_G$ for every maximal subgroup V of the generalized Fitting subgroup \(F^{*}(G)\) $F_{}^{\ast }(G)$ of G; a finite soluble group G is a PST-group (that is, Sylow permutability is a transitive relation on G) if and only if G has a normal subgroup D such that G/D is nilpotent and D avoids every chief factor of G between \(V^{G}\) $V^G$ and \(V_{G}\) $V_G$ for every subnormal subgroup A of G.